David McLaren

Elena Celledoni, David McLaren, Brynjulf Owren et al.

Given a quadratic vector field on \mathbb{R}^n possessing a quadratic first integral depending on two of the independent variables, we give a constructive proof that Kahan's discretization method exactly preserves a nearby modifed integral. Building on this result, we present a family of integrable quadratic vector fields (including the Euler top) whose Kahan discretization is a novel 10-parameter family of integrable maps.

Elena Celledoni, Robert I. McLachlan, David I. McLaren et al.

Applying Kahan's discretization to the reduced Nahm equations, we obtain two classes of integrable mappings.

Philipp Bader, David I McLaren, G. R. W. Quispel et al.

It is a classical theorem of Liouville that Hamiltonian systems preserve volume in phase space. Any symplectic Runge-Kutta method will respect this property for such systems, but it has been shown that no B-Series method can be volume preserving for all volume preserving vector fields (BIT 47 (2007) 351-378 and IMA J. Numer. Anal. 27 (2007) 381-405). In this paper we show that despite this result, symplectic Runge-Kutta methods can be volume preserving for a much larger class of vector fields than Hamiltonian systems, and discuss how some Runge-Kutta methods can preserve a modified measure exactly.